Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres. The spheres are suspended by non-conducting thread. Each sphere has a mass of $3.2 \times 10^{-3} \text{ kg}$ and a radius of 20 mm. The distance $d$ is 40 mm. The capacitance of a sphere of radius $r$ is $4\pi \epsilon_{0} r$. Each sphere is charged by connecting it briefly to the positive terminal of a high-voltage supply, the other terminal of which is at 0 V. After this has been done the charge on each sphere is 52 nC. Calculate the potential of one of the spheres. The charged spheres in Figure 3 are at equilibrium. Draw labelled arrows on Figure 3 to show the forces on sphere B. Suggest a solution to one problem involved in the measurement of $d$ in Figure 3. Show that the magnitude of the electrostatic force on each sphere is about $4 \times 10^{-3} \text{ N}$. A student measures the angle $\theta$ when the apparatus in Figure 3 is at equilibrium. The student records $\theta = 7^{\circ}$. Discuss whether this measurement is consistent with the other data in this investigation. The student says that the gravitational force between the two spheres has no significant effect on the angle that the spheres are in equilibrium. Deduce with a calculation whether this statement is valid.

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Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres - AQA - A-Level Physics - Question 4 - 2019 - Paper 2

Question 4

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Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres. The spheres are suspended by non-conducting ... show full transcript

Worked Solution & Example Answer:Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres - AQA - A-Level Physics - Question 4 - 2019 - Paper 2

Step 1

Calculate the potential of one of the spheres.

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Answer

To calculate the potential ( V) of one sphere, we can use the formula:

C = 4\pi \epsilon_{0} r

Substituting in the known values:

Radius ( $r$ ) = 20 mm = 0.02 m
Capacitance ( C) = $4\pi (8.85 \times 10^{-12} F/m) (0.02 m)$

Calculating:

C = 4\pi (8.85 \times 10^{-12}) (0.02) = 2.22 \times 10^{-12} F

Using charge ( Q) which is 52 nC = $52 \times 10^{-9} C$ we find the potential:

V = \frac{Q}{C} = \frac{52 \times 10^{-9}}{2.22 \times 10^{-12}} \approx 23,000 V

Step 2

Draw labelled arrows on Figure 3 to show the forces on sphere B.

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Answer

Forwards the drawing on sphere B, the following forces should be represented:

The gravitational force acting downwards (label it as $F_g$ ).
The electrostatic repulsive force acting horizontally away from sphere A (label it as $F_e$ ).
The tension in the thread acting diagonally upwards towards the point of support (label it as $F_t$ ).

Step 3

Suggest a solution to one problem involved in the measurement of d in Figure 3.

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Answer

One problem in measuring $d$ is ensuring that the measurement is accurate as the threads can stretch and the spheres can oscillate.

To improve accuracy, a more stable measuring device, such as a ruler with a fixed angle from the point of equilibrium, could be used.

Step 4

Show that the magnitude of the electrostatic force on each sphere is about 4 × 10−3 N.

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Answer

Using Coulomb's law, the electrostatic force ( F) between two charges is given by:

F = k \frac{|q_1 q_2|}{d^2}

Where:

$k = 8.99 \times 10^9 ext{ N m}^2/ ext{C}^2$
$q_1 = q_2 = 52 \times 10^{-9} C$
$d = 0.04 m$

Thus,

F = 8.99 \times 10^9 \frac{(52 \times 10^{-9})^2}{(0.04)^2} \approx 4 \times 10^{-3} N

Step 5

Discuss whether this measurement is consistent with the other data in this investigation.

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Answer

Considering that the calculated forces and potential values are significantly greater than gravitational forces, this indicates that the electrostatic forces play a dominant role in equilibrium. Therefore, the angle measurement of $\theta = 7^{\circ}$ is consistent with the other data, given the strong repulsive force evident in the calculations.

Step 6

Deduce with a calculation whether this statement is valid.

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Answer

To assess whether the gravitational force has no significant effect, we can compare the gravitational force with the electrostatic force.

The gravitational force ( $F_g$ ) can be calculated as:

F_g = mg = (3.2 \times 10^{-3} kg)(9.81 m/s^2) \approx 0.0314 N

From the previous calculations, the electrostatic force ( $F_e$ ) is approximately $4 \times 10^{-3} N$ .

Clearly, $F_g$ is larger than $F_e$ , indicating that the gravitational force does indeed have a significant effect on the equilibrium of the spheres.

Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres - AQA - A-Level Physics - Question 4 - 2019 - Paper 2

Worked Solution & Example Answer:Figure 3 shows an arrangement used to investigate the repulsive forces between two identical charged conducting spheres - AQA - A-Level Physics - Question 4 - 2019 - Paper 2

Calculate the potential of one of the spheres.

Draw labelled arrows on Figure 3 to show the forces on sphere B.

Suggest a solution to one problem involved in the measurement of d in Figure 3.

Show that the magnitude of the electrostatic force on each sphere is about 4 × 10−3 N.

Discuss whether this measurement is consistent with the other data in this investigation.

Deduce with a calculation whether this statement is valid.

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